**5.5. Eight-beam case**

**Figure 10.** [*E*(*x*)] and [*S*(*x*)] (*x* ∈ {*a*, *b*}) are experimentally obtained and computer-simulated eight-beam X-ray pinhole topographs with an incidence of (*a*) horizontal-linearly and (*b*) vertical-linearly polarized X-rays whose photon energy was 18.245 keV. The exposure time for [*E*(*x*)] was 240 s.

Figure 10[*E*(*a*)] and 10[*S*(*a*)] are eight-beam X-ray pinhole topographs experimentally obtained and computer-simulated, respectively, with an incidence of horizontal-linearly

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section are six-beam pinhole topographs whose Borrmann pyramid is not a regular hexagonal

Such six-beam pinhole topographs experimentally obtained and computer-simulated are shown in Figure 9. Figures 9[*E*(*b*)] and 9[*S*(*b*)] are enlargements of 2 6 4- and 0 6 6-reflected X-ray images from Figures 9[*E*(*a*)] and 9[*S*(*a*)]. Knife-edge patterns [*KEL*(1) and *KEL*(2)] indicated by arrows in Figure 9[*S*(*b*)] are found also in Figure 9[*E*(*b*)]. Circular patterns that were found in the central part of the six-beam pinhole topographs [14, 16, 17] cannot be found

in the present case. A 'heart-shaped' pattern (*HSP*) is found also in Figure 9[*E*(*b*)].

**Figure 10.** [*E*(*x*)] and [*S*(*x*)] (*x* ∈ {*a*, *b*}) are experimentally obtained and computer-simulated

polarized X-rays whose photon energy was 18.245 keV. The exposure time for [*E*(*x*)] was 240 s.

eight-beam X-ray pinhole topographs with an incidence of (*a*) horizontal-linearly and (*b*) vertical-linearly

Figure 10[*E*(*a*)] and 10[*S*(*a*)] are eight-beam X-ray pinhole topographs experimentally obtained and computer-simulated, respectively, with an incidence of horizontal-linearly

pyramid.

**5.5. Eight-beam case**

**Figure 11.** [*E*(*x*)] and [*S*(*x*)] (*x* ∈ {*a*, *b*}) are enlargements of 0 0 0-forward-diffracted X-ray images in Figures 10 [*E*(*x*)] and 10 [*S*(*x*)].

polarized X-rays. Figure 10[*E*(*b*)] and 10[*S*(*b*)] were obtained with an incidence of vertical-linearly polarized X-rays. Figure 11[*E*(*x*)] and 11[*S*(*x*)] (*x* ∈ {*a*, *b*}) are enlargements of 0 0 0-forward-diffracted X-ray images from Figures 10[*E*(*x*)] and 10[*S*(*x*)], respectively.

In Figure 11[*S*(*a*)], A 'harp-shaped' pattern (*HpSP*), knife-edge line (*KEL*), 'hook-shaped' pattern (*HkSP*), 'Y-shaped' pattern (*YSP*) and 'nail-shaped' patterns are indicated by arrows. All these characteristic patterns are observed also in Figure 11[*E*(*a*)]. *NSP* is also observed in Figures 11[*E*(*b*)] and 11[*S*(*b*)]. However, *HpSP* in Figures 11[*E*(*b*)] and 11[*S*(*b*)] are rather fainter compared with Figures 11[*E*(*a*)] and 11[*S*(*a*)].

#### 18 Will-be-set-by-IN-TECH 84 Recent Advances in Crystallography

#### **5.6. Twelve-beam case**

**Figure 12.** [*E*(*a*)] and [*S*(*a*)] are experimentally obtained and computer-simulated twelve-beam X-ray pinhole topographs with an incidence of horizontal-linearly polarized X-rays whose photon energy was 22.0 keV. [*E*(*b*)] and [*S*(*b*)] are enlargements of 2 4 2-reflected X-ray images in [*E*(*a*)] and [*S*(*a*)]. The exposure time for [*E*(*a*)] and [*E*(*b*)] was 300 s.

Twelve is the largest number of *n* for the *n*-beam T-T equation (14) that restricts a condition that *n* reciprocal lattice nodes should ride on a circle in reciprocal space. Figures 12[*E*(*a*)] and 12[*S*(*a*)] are experimentally obtained and computer-simulated tweleve-beam pinhole topographs. Figures 12[*E*(*b*)] and 12[*S*(*b*)] are enlargements of 2 4 2-reflected X-ray images from Figures 12[*E*(*a*)] and 12[*S*(*a*)].

A very bright region (*VBR*), 'V-shaped' pattern (*VSP*), central circle (*CC*) and 'U-shaped' pattern indicated by arrows in Figure 12[*S*(*b*)] are found also in Figure 12[*E*(*b*)].

## **6. Summary**

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**Figure 12.** [*E*(*a*)] and [*S*(*a*)] are experimentally obtained and computer-simulated twelve-beam X-ray pinhole topographs with an incidence of horizontal-linearly polarized X-rays whose photon energy was 22.0 keV. [*E*(*b*)] and [*S*(*b*)] are enlargements of 2 4 2-reflected X-ray images in [*E*(*a*)] and [*S*(*a*)]. The

Twelve is the largest number of *n* for the *n*-beam T-T equation (14) that restricts a condition that *n* reciprocal lattice nodes should ride on a circle in reciprocal space. Figures 12[*E*(*a*)] and 12[*S*(*a*)] are experimentally obtained and computer-simulated tweleve-beam pinhole topographs. Figures 12[*E*(*b*)] and 12[*S*(*b*)] are enlargements of 2 4 2-reflected X-ray images

A very bright region (*VBR*), 'V-shaped' pattern (*VSP*), central circle (*CC*) and 'U-shaped'

pattern indicated by arrows in Figure 12[*S*(*b*)] are found also in Figure 12[*E*(*b*)].

exposure time for [*E*(*a*)] and [*E*(*b*)] was 300 s.

from Figures 12[*E*(*a*)] and 12[*S*(*a*)].

**5.6. Twelve-beam case**

The *n*-beam (*n* ∈ {3, 4, 5, 6, 8, 12}) Takagi-Taupin equation and computer algorithm to solve it were verified from excellent agreements between experimentally obtained and computer-simulated three-, four-, five-, six-, eight- and twelve-beam pinhole topographs.

The equivalence between the E-L and T-T formulations of the *n*-beam X-ray dynamical diffraction theory, which has been implicitly recognized for two-beam case theory, was explicitly described in the present chapter. Whereas the former theory can be calculated by solving an eigenvalue-eigenvector problem, the latter can be calculated by solving a partial differential equation. This equivalence is very similar to that between the Heisenberg and Schrödinger pictures of quantum mechanics and is very important.

Whereas this chapter has been described with focusing on the *n*-beam case that *n* ∈ {3, 4, 5, 6, 8, 12}, the *n*-beam X-ray dynamical diffraction theory applicable to the case of arbitrary number of *n*, which is effective and important for solving the phase problem in protein crystal structure analysis, will be described elsewhere. In the case of protein crystallography, the situation that arbitrary number of reciprocal lattice nodes are very close to the surface of the Ewald sphere, cannot be avoided. In protein crystallography, the *n*-beam X-ray dynamical diffraction theory for arbitrary number of *n* is necessary.
